Properties

Label 6005.2.a.e
Level $6005$
Weight $2$
Character orbit 6005.a
Self dual yes
Analytic conductor $47.950$
Analytic rank $1$
Dimension $88$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6005,2,Mod(1,6005)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6005, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6005.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6005 = 5 \cdot 1201 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6005.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(47.9501664138\)
Analytic rank: \(1\)
Dimension: \(88\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 88 q - 14 q^{2} - 34 q^{3} + 66 q^{4} + 88 q^{5} - q^{6} - 35 q^{7} - 39 q^{8} + 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 88 q - 14 q^{2} - 34 q^{3} + 66 q^{4} + 88 q^{5} - q^{6} - 35 q^{7} - 39 q^{8} + 72 q^{9} - 14 q^{10} - 26 q^{11} - 64 q^{12} - 31 q^{13} - 17 q^{14} - 34 q^{15} + 34 q^{16} - 31 q^{17} - 42 q^{18} - 56 q^{19} + 66 q^{20} - q^{21} - 49 q^{22} - 74 q^{23} - 3 q^{24} + 88 q^{25} - q^{26} - 130 q^{27} - 57 q^{28} - 6 q^{29} - q^{30} - 37 q^{31} - 87 q^{32} - 43 q^{33} - 35 q^{34} - 35 q^{35} + 53 q^{36} - 67 q^{37} - 40 q^{38} - 21 q^{39} - 39 q^{40} + 2 q^{41} - 15 q^{42} - 136 q^{43} - 15 q^{44} + 72 q^{45} - 16 q^{46} - 139 q^{47} - 71 q^{48} + 41 q^{49} - 14 q^{50} - 71 q^{51} - 71 q^{52} - 75 q^{53} + 26 q^{54} - 26 q^{55} - 22 q^{56} - 34 q^{57} - 65 q^{58} - 41 q^{59} - 64 q^{60} - 11 q^{61} - 30 q^{62} - 114 q^{63} - 33 q^{64} - 31 q^{65} + 24 q^{66} - 209 q^{67} - 42 q^{68} - 22 q^{69} - 17 q^{70} - 43 q^{71} - 80 q^{72} - 50 q^{73} + 9 q^{74} - 34 q^{75} - 62 q^{76} - 49 q^{77} - 19 q^{78} - 77 q^{79} + 34 q^{80} + 72 q^{81} - 107 q^{82} - 113 q^{83} + 19 q^{84} - 31 q^{85} + 14 q^{86} - 87 q^{87} - 107 q^{88} - 5 q^{89} - 42 q^{90} - 159 q^{91} - 100 q^{92} - 82 q^{93} - 31 q^{94} - 56 q^{95} + 58 q^{96} - 105 q^{97} - 29 q^{98} - 68 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.76833 −0.546396 5.66368 1.00000 1.51261 −1.48260 −10.1423 −2.70145 −2.76833
1.2 −2.70073 −2.65354 5.29393 1.00000 7.16648 −4.13839 −8.89601 4.04126 −2.70073
1.3 −2.69844 0.716329 5.28159 1.00000 −1.93297 1.67359 −8.85517 −2.48687 −2.69844
1.4 −2.69778 −3.22929 5.27799 1.00000 8.71190 2.27707 −8.84328 7.42831 −2.69778
1.5 −2.63715 2.33888 4.95457 1.00000 −6.16798 −0.829927 −7.79166 2.47036 −2.63715
1.6 −2.56191 2.09483 4.56337 1.00000 −5.36676 −0.643691 −6.56712 1.38832 −2.56191
1.7 −2.49547 −2.51949 4.22735 1.00000 6.28731 2.97292 −5.55828 3.34784 −2.49547
1.8 −2.45453 −0.943055 4.02473 1.00000 2.31476 −0.315218 −4.96977 −2.11065 −2.45453
1.9 −2.32346 1.00159 3.39848 1.00000 −2.32717 3.48702 −3.24933 −1.99681 −2.32346
1.10 −2.28766 −2.82727 3.23337 1.00000 6.46782 −3.61501 −2.82152 4.99347 −2.28766
1.11 −2.26517 2.21267 3.13101 1.00000 −5.01209 −3.04532 −2.56193 1.89592 −2.26517
1.12 −2.25928 −0.374349 3.10434 1.00000 0.845758 −3.16273 −2.49501 −2.85986 −2.25928
1.13 −2.21323 −3.30657 2.89841 1.00000 7.31821 −1.75850 −1.98838 7.93340 −2.21323
1.14 −2.21193 0.112500 2.89262 1.00000 −0.248842 −5.20479 −1.97441 −2.98734 −2.21193
1.15 −2.17289 −1.65848 2.72145 1.00000 3.60369 5.14399 −1.56762 −0.249442 −2.17289
1.16 −2.07691 −0.413168 2.31355 1.00000 0.858111 2.69878 −0.651210 −2.82929 −2.07691
1.17 −1.97194 2.67165 1.88854 1.00000 −5.26832 −0.758910 0.219801 4.13769 −1.97194
1.18 −1.97101 −1.60435 1.88489 1.00000 3.16220 2.33314 0.226875 −0.426059 −1.97101
1.19 −1.93565 −2.84784 1.74674 1.00000 5.51242 −0.215804 0.490231 5.11019 −1.93565
1.20 −1.89204 0.756696 1.57982 1.00000 −1.43170 1.91014 0.794996 −2.42741 −1.89204
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.88
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(1201\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6005.2.a.e 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6005.2.a.e 88 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{88} + 14 T_{2}^{87} - 23 T_{2}^{86} - 1205 T_{2}^{85} - 2777 T_{2}^{84} + 47070 T_{2}^{83} + 207654 T_{2}^{82} - 1068382 T_{2}^{81} - 7279036 T_{2}^{80} + 14221060 T_{2}^{79} + 166802742 T_{2}^{78} + \cdots + 32 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6005))\). Copy content Toggle raw display